Asymptotic tracts of harmonic functions. III (Q1922838)

Summary: [For Parts I and II, see Ann. Acad. Sci. Fenn., Ser. A I 11, 215-232 (1986; Zbl 0594.31002) and Proc. Edinb. Math. Soc., II. Ser. 38, No. 1, 35-52 (1995; Zbl 0818.31002).] A tract (or asymptotic tract) of a real function \(u\) harmonic and nonconstant in the complex plane \(\mathbb{C}\) is one of the \(n_c\) components of the set \(\{z: u(z)\neq c\}\), and the order of a tract is the number of non-homotopic curves from any given point to \(\infty\) in the tract. The authors prove that if \(u(z)\) is an entire harmonic polynomial of degree \(n\), if the critical points of any of its analytic completions \(f\) lie on the level sets \(\tau_j= \{z: u(z)=c_j\}\), where \(1\leq j\leq p\) and \(p\leq n-1\), and if the total order of all the critical points of \(f\) on \(\tau_j\) is denoted by \(\sigma_j\), then \[ \{n_c: c\in\mathbb{R}\}= \{n+1\}\cup \{n+1+ \sigma_j: 1\leq j\leq p\}. \]